Several months ago I had a post (rational credences–still private!) about Rachael Briggs’ paper Distorted Reflection and Anna Mahtani’s reply to it. Unfortunately, I had read Mahtani’s paper, which capitalizes on a single unfortunate sentence on the last page of Briggs’ paper, first. The sentence is this: “I’ll adopt Lewis’s assumption that the agent stands no chance of mistaking her evidence, so that if the agent learns E, then all the suppositional worlds must be ones where E is true.” Having now read the papers more carefully…granted still not so carefully…Briggs’s paper is great. One of the best I’ve read. Mahtani’s paper meanwhile, in focusing on that one sentence and not the overall thrust of Briggs’ paper, is something of an uncharitable hatchet job–all it shows is that that one sentence doesn’t characterize what Briggs is consistently doing throughout the rest of her paper.
All Briggs was really saying here was that any agent who fails to (conditionalize actually) (actually regards as fair) a sequence of bets yielding a net loss in every possible world, yet there are cases of agents who (actually fail to reflect) but–despite being Dutch Booked–do not (actually regard as fair) a sequence of bets yielding a net loss in every possible world. For them, there are worlds where the bets actually accepted make money. (That said–it’s a Dutch Book after all–in those worlds they do go on to accept a further bet that loses whatever they gained and then some.) Very nice insight.
One might think there is a further distinction to be made…if a violator of reflection condones, in moments of lucidity, the counterfactual violations of conditionalization that lead to the reflection violation, they should be held accountable. Although there is a sense in which they may be diachronically coherent…their credences over time may mirror epistemic frequencies precisely after all…there is a finer sense in which they are coherent by virtue of luck. (Even a random updater would turn out to be coherent some of the time.)
There’s a stronger sense of coherence, to have which implies that–so long as you are able to maintain lucidity, of course–you’ll be coherent almost surely. As far as Dutch Books go, the lucidity requirement is confusing…after all, to lose one’s lucidity may just be what constitutes irrationality. So perhaps the real issue is one of consent. On such a view a legitimate Dutch Book is a seduction establishing incoherence, whereas an illegitimate Dutch Book is a rape establishing nothing of the kind.
To make this point rather tastelessly:
Rose’s credence in “Bill’s a worthy mate” conditional on Bill giving her flowers is near zero but she knows that if Bill actually gives her flowers then her credence in “Bill’s a worthy mate” will rise to nearly 1. Quai’s credence in “Bill’s a worthy mate” conditional on Bill slipping a lude into her drink without her consent is near zero, but she knows that if Bill actually slips a lude into her drink without her consent, her credence in “Bill’s a worthy mate” will rise to nearly 1…for all her ensuing resistance will establish to the contrary, at any rate. Both fail reflection, but I would be inclined to maintain that Rose is irrational, whereas Quai is not. It’s not irrational to be a potental victim of a crime. It is irrational to be a potential victim of an ill advised series of gambles.
But the terrain here is trickier than that…one could maintain that Rose’s case (as culpability for her credences goes) is the same as Quai’s…it’s not her fault that she’s affected as she is by flowers. Flowers have this scent that acts very much like a drug for Rose….blah blah blah. Obviously now we’ve gotten out of the realm of philosophy of probability, where I have no special claim to know what I am talking about.
That’s sort of my point about Briggs’s paper, though…I think she took it just about as far as you can take it from within the confines of philosophy of probability. Yes, she appeared to leave something out…she let many reflection violators off the hook. She did this by killing all of the Dutch Books for reflection. Some of the Dutch Books for reflection are probably okay. Namely, those in which all bets are agreed to with lucidity or with culpability for non-lucidity. Or something like that, but there’s a problem with this approach. The only way I see to define lucidity here is that the betting behavior should agree with that planned earlier in some sort of original moment of lucidity. But that makes the whole discussion sort of moot….surely no one would devise, in a moment of lucidity, a credal scheme susceptible to reflection violations. The whole point of a reflection violation is that it’s supposed to be a violation of diachronic norms….rationality over time. You destroy that idea if you only accept Dutch Books that would be sanctioned in the original moment of lucidity, for now you are just talking about synchronic rationality.
I still think that the best way to define diachronic coherence is the way I did in my last post. But I now recognize that my stronger definition…coherence of a credal scheme…isn’t really a diachronic notion. And probably isn’t very useful. I also am a realist. Philosophers are not going to like the way I defined diachronic coherence in the first place…because they aren’t comfortable with the frequency theory, for example. Dutch Books seem to be something that philosophers can live with, and though they are clunkier, there isn’t anything wrong with them. Within the constraints of analyzing credal coherence from the confines of Dutch Book lore, I would say Briggs absolutely killed it. (Caveat: she says that someone called Jonathan Weisberg had similar results as these appear in 2007. I haven’t looked at his paper yet. Maybe he deserves a slice of the credit for what’s here. I shall look soon, anyway.)
So are there ways to put reflection violators who aren’t coerced back on the hook? I’m sure there are. I just doubt now that this can be done from within the confines of “philosophy of probablity”. What makes the question so murky is that you want to put the hammer on voluntary, but not involuntary, deviations from what I have described as a coherent credal scheme. But who would deviate voluntarily? No one. Textbook cases of incoherence just are involuntary. It’s the same with adopting a credence function in accord with the probability axioms….everyone wants to but no one does. We don’t have the computation resources. Our violations are involuntary…that doesn’t make them coherent!
The alternative is to let all reflection violators off the hook…unless and until they violate conditionalization. I believe that’s the spirit of Briggs’s paper. Is it so bad? I’m getting used to it. After all, it’s not so much coherence as accuracy we’re after in the adoption of our credence functions. Sure, some self-doubters might get lucky and remain coherent in the short term. But if their self-doubts are well founded, their incoherence will eventually out and, if they aren’t, their accuracy will suffer for having them. This is the credal analog of what Briggs was talking about when she said “there is a sense in which self-doubting agents can’t be right about everything.” Where beliefs are concerned, this comes to light right away. Where credences are concerned, you may need to wait long enough for “the law of large numbers to kick in” as they say. Self-doubters are either wrong to self-doubt, in which case accuracy will suffer in the long run, or they’re right to self doubt, and coherence (hence accuracy…the only reason to be coherent anyway) will suffer in the long run.
Wait…did I just put the reflection violators back on the hook from within the confines of philosophy of probability? Damn. Well, it sounded like it would be hard at the time.
Oh and obviously if you are coerced, well, your coherence/accuracy may suffer then, too. We’re all on that particular hook. Because while crime may not pay, it sure as hell hurts.